In a previous paper, we gave a correspondence between certain exact solutions to a (2 + 1)-dimensional integrable Chiral Model and holomorphic bundles on a compact surface. In this paper, we use algebraic geometry to derive a closed-form expression for those solutions and show by way of examples how the algebraic data which parametrise the solution space dictates the behaviour of the solutions.

The problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed $F-$, $\text{DF}$-power series spaces, i.e. the spaces of the following kind

$$G(\lambda ,a)=\underset{p\to \infty }{\mathop{\lim }}\,\text{proj}\left( \underset{q\to \infty }{\mathop{\lim }}\,\text{ind}\left( {{\ell }_{1}}\left( {{a}_{i}}\left( p,q \right) \right) \right) \right),$$

where ${{a}_{i}}(p,\,q)\,=\,\exp \left( \left( p\,-\,{{\lambda }_{i}}q \right){{a}_{i}} \right),\,p,\,q\,\in \,\mathbb{N},\,\text{and}\,\lambda \,\text{=}\,{{\left( {{\lambda }_{i}} \right)}_{i\in \mathbb{N}}},\,a=\,{{({{a}_{i}})}_{i\in \mathbb{N}}}$ are some sequences of positive numbers. These spaces, up to isomorphisms, are basis subspaces of tensor products of power series spaces of $F-$ and $\text{DF}$-types, respectively. The m-rectangle characteristic $\mu _{m}^{\lambda ,a}\left( \delta ,\,\varepsilon ;\,\tau ,\,t \right),\,m\,\in \,\mathbb{N}$ of the space $G\left( \text{ }\!\!\lambda\!\!\text{ ,}\,a \right)$ is defined as the number of members of the sequence ${{({{\lambda }_{i}},{{a}_{i}})}_{i\in \mathbb{N}}}$ which are contained in the union of m rectangles ${{P}_{k}}\,=\,\left( {{\delta }_{k}},\,{{\varepsilon }_{k}} \right]\,\times \,\left( {{\tau }_{k}},\,{{t}_{k}} \right]$ , $k\,=\,1,2,\ldots ,m$. It is shown that each m-rectangle characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invariants, which combine some version of the classical approximative dimensions (Kolmogorov, Pełczynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis).

We introduce a distributional Ricci curvature on complete smooth manifolds with Lipschitz continuous metrics. Under an assumption on the volume growth of geodesics balls, we obtain a gradient estimate for weakly harmonic functions if the distributional Ricci curvature is bounded below.

In this paper, we solve the isomorphism problem for metacirculant graphs of order $pq$ that are not circulant. To solve this problem, we first extend Babai’s characterization of the $\text{CI}$-property to non-Cayley vertex-transitive hypergraphs. Additionally, we find a simple characterization of metacirculant Cayley graphs of order $pq$, and exactly determine the full isomorphism classes of circulant graphs of order $pq$.

Abelian closed subgroups of the Galois group of the pythagorean closure of a formally real field are described by means of the inertia group of suitable valuation rings.

Let $X$ be a smooth projective surface over the complex number field and let $L$ be a nef-big divisor on $X$. Here we consider the following conjecture; If the Kodaira dimension $\kappa (X)\ge 0$, then ${{K}_{X}}L\,\ge \,2q(X)\,-\,4$, where $q\left( X \right)$ is the irregularity of $X$. In this paper, we prove that this conjecture is true if (1) the case in which $\kappa (X)=0$ or 1, (2) the case in which $\kappa (X)=2$ and ${{h}^{0}}(L)\,\ge \,2$ , or (3) the case in which $\kappa (X)=2$, $X$ is minimal, ${{h}^{0}}(L)\,=\,1$ , and $L$ satisfies some conditions.

We consider the analogue of the $\wedge (p)$―problem for subsets of the Legendre polynomials or more general ultraspherical polynomials. We obtain the “best possible” result that if $2\,<\,p\,<\,4$ then a random subset of $N$ Legendre polynomials of size ${{N}^{4/p-1}}$ spans an Hilbertian subspace. We also answer a question of König concerning the structure of the space of polynomials of degree $n$ in various weighted ${{L}_{p}}$-spaces.

For an arbitrary finite Galois $p$-extension $L/K$ of ${{\mathbb{Z}}_{p}}$-cyclotomic number fields of $\text{CM}$-type with Galois group $G=\text{Gal}(L/K)$ such that the Iwasawa invariants $\mu _{K}^{-},\,\mu _{L}^{-}$ are zero, we obtain unconditionally and explicitly the Galois module structure of $C_{L}^{-}\,(p)$, the minus part of the $p$-subgroup of the class group of $L$. For an arbitrary finite Galois $p$-extension $L/K$ of algebraic function fields of one variable over an algebraically closed field $k$ of characteristic $p$ as its exact field of constants with Galois group $G=\text{Gal}(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$-torsion part of the Jacobian variety ${{J}_{L}}(p)$ associated to $L/k$.

We obtain necessary and sufficient conditions for mean convergence of Lagrange interpolation at zeros of orthogonal polynomials for weights on [-1, 1], such as

$$w(x)\,=\,\exp \left( -{{\left( 1-{{x}^{2}} \right)}^{-\alpha }} \right),\,\alpha >0$$

or

$$w(x)=\exp \left( -{{\exp }_{k}}{{\left( 1-{{x}^{2}} \right)}^{-\alpha }} \right),k\ge 1,\alpha >0,$$

where ${{\exp }_{k}}=\exp \left( \exp \left( \cdot \cdot \cdot \exp (\,)\cdot \cdot \cdot \right) \right)$ denotes the $k$-th iterated exponential.

Turbiner’s conjecture posits that a Lie-algebraic Hamiltonian operator whose domain is a subset of the Euclidean plane admits a separation of variables. A proof of this conjecture is given in those cases where the generating Lie-algebra acts imprimitively. The general form of the conjecture is false. A counter-example is given based on the trigonometric Olshanetsky-Perelomov potential corresponding to the ${{A}_{2}}$ root system.

Nous montrons que l’invariant de Hasse-Witt de la forme de Killing d’une algèbre de Lie semi-simple $L$ s’exprime à l’aide de l’invariant de Tits de la représentation irréductible de $L$ de poids dominant $\rho \,=\,\frac{1}{2}$ (somme des racines positives), et des invariants associés au groupe des symétries du diagramme de Dynkin de $L$.